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Find the sum of the infinite geometric series.\newline5+53+59+527+…5 + \frac{5}{3} + \frac{5}{9} + \frac{5}{27} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_

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Q. Find the sum of the infinite geometric series.\newline5+53+59+527+…5 + \frac{5}{3} + \frac{5}{9} + \frac{5}{27} + \dots\newlineWrite your answer as an integer or a fraction in simplest form.\newline__\_\_
  1. Divide Total Amount: To solve this problem, we need to divide the total amount of electrical tape needed by the amount of tape on each roll.\newlineCalculation: 8,000 cm÷2,000 cm/roll8,000 \, \text{cm} \div 2,000 \, \text{cm/roll}
  2. Calculate Number of Rolls: Evaluating the division gives us the number of rolls needed. 8,000 cm÷2,000 cm/roll=4 rolls8,000 \, \text{cm} \div 2,000 \, \text{cm}/\text{roll} = 4 \, \text{rolls}
  3. Identify First Term and Ratio: First, identify the first term aa and the common ratio rr of the geometric series.\newlineThe first term aa is 55, and the common ratio rr is 13\frac{1}{3}, since each term is 13\frac{1}{3} of the previous term.
  4. Find Sum Formula: The sum of an infinite geometric series can be found using the formula S=a1−rS = \frac{a}{1 - r}, where SS is the sum, aa is the first term, and rr is the common ratio.\newlineHere, a=5a = 5 and r=13r = \frac{1}{3}.
  5. Substitute Values: Substitute the values of aa and rr into the formula to find the sum.S=5(1−13)S = \frac{5}{(1 - \frac{1}{3})}
  6. Calculate Denominator: Calculate the denominator of the fraction: 1−13=231 - \frac{1}{3} = \frac{2}{3}.
  7. Divide by Fraction: Now, divide the first term by the result from the previous step to find the sum of the series. S=523S = \frac{5}{\frac{2}{3}}
  8. Perform Multiplication: To divide by a fraction, multiply by its reciprocal.\newlineS=5×(32)S = 5 \times \left(\frac{3}{2}\right)
  9. Perform Multiplication: To divide by a fraction, multiply by its reciprocal.\newlineS=5×(32)S = 5 \times \left(\frac{3}{2}\right)Perform the multiplication to find the sum.\newlineS=152S = \frac{15}{2} or 7.57.5\newlineHowever, since the question prompt asks for an integer or a fraction in simplest form, we leave the answer as a fraction.

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